### Algebraic Geometry 4: A short note on Projective Varieties

What is a variety? It is the set of common zeroes for a set of polynomials. For example, for the set of polynomials $\{x+y,x-y\}\in\Bbb{R}[x,y]$, the variety is $(0,0)$.

Now what is a projective variety? Simply put, it is the common set of zeroes of polynomials in which a one-dimensional subspace is effectively considered one point. Hence, for the variety to be well-defined, if one point of a one-dimensional subspace satisies the variety, every point of the one-dimensional subspace has to satisfy that variety. Confused?

Take the polynomial $x+y+z\in\Bbb{C}[x,y,z]$. The point $(1,-1,0)$ satisfies this polynomial. Now note that the points $\lambda(1,-1,0)$ also satisfy this polynomial for every $\lambda\in\Bbb{C}$. Hence this is a projective variety. Now take $x+y+z-1\in\Bbb{C}[x,y,z]$. Here $\lambda(1,0,0)$ satisfies the polynomial for only $\lambda=1$. Hence, this is not a projective variety.

But why? Why would you want to consider a whole line as one point? When you watch the world from your little nest, every line running along your ine of sight becomes a point. Hence, athough it may be a line in “reality” (whatever this means), for you it is a point. This is the origin of projective geometry, although things have gotten sightly complicated since then.